Linear Theory of Abstract Functional Differential Equations of Retarded Type

The purpose of this paper is to provide an extension of the linear theory of functional differential equations of retarded type to abstract equations. Such equations include examples borrowed from population dynamics to which the theory applies. An application will be given elsewhere. Our main effort in this work consists in providing a suitable extension of the formal adjoint equation and the formal duality. The solutions of the linear autonomous retarded functional differential equation x′(t) = L(xt), (1) where L is a bounded linear operator mapping the space C([−r, 0]; E) into the Banach space E, define a strongly continuous translation semigroup. We show the existence of a direct sum decomposition of C([−r, 0]; E) into two subspaces which are semigroup invariants. The flow induced by the solutions of Eq. (1) can be interpreted as the flow induced by an ordinary differential equation in a finite-dimensional space. We explicitly characterize this decomposition by an orthogonality relation associated to a certain definition of formal duality. The existence of an integral representation for the operator L leads to an equation formally adjoint to (1) characterizing the projection operator defined by the above decomposition of C([−r, 0]; E).